2. Theoretical background
2.1. Learning mathematics in context Contexts are recognized as important levers for mathematics learning because they offer various
opportunities for students to learn mathematics. The use of contexts reduces students’ perception of mathematics as a remote body of knowledge Boaler, 1993b, and by means of contexts students can
develop a better insight about the usefulness of mathematics for solving daily-life problems De Lange, 1987. Another benefit of contexts is that they provide students with strategies to solve
mathematical problems Van den Heuvel-Panhuizen, 1996. When solving a context-based problem, students might connect the situation of the problem to their experiences. As a result, students might
use not only formal mathematical procedures, but also informal strategies, such as using repeated subtraction instead of a formal digit-based long division. In the teaching and learning process,
students’ daily experiences and informal strategies can also be used as a starting point to introduce mathematics concepts. For example, covering a floor with squared tiles can be used as the starting
point to discuss the formula for the area of a rectangle. In this way, contexts support the development of students’ mathematical understanding De Lange, 1987; Gravemeijer Doorman, 1999; Van den
Heuvel-Panhuizen, 1996. In mathematics education, the use of contexts can imply different types of contexts.
According to Van den Heuvel-Panhuizen 2005 contexts may refer to real-world settings, fantasy situations or even to the formal world of mathematics. This is a wide interpretation of
context in which contexts are not restricted to real-world settings. What important is that contexts create situations for students that are experienced as real and related to their common-
sense understanding. In addition, a crucial characteristic of a context for learning mathematics is that there are possibilities for mathematization. A context should provide information that can be
organized mathematically and should offer opportunities for students to work within the context by using their pre-existing knowledge and experiences Van den Heuvel-Panhuizen, 2005.
The PISA study also uses a broad interpretation of context, defining it as a specific setting within a ‘situation’ which includes all detailed elements used to formulate the problem OECD,
2003b, p. 32. In this definition, ‘situation’ refers to the part of the students’ world in which the tasks are placed. This includes personal, educationaloccupational, public, and scientific situation
types. As well as Van den Heuvel-Panhuizen 2005, the PISA researchers also see that a formal mathematics setting can be seen as a context. Such context is called an ‘intra-mathematical
context’ OECD, 2003b, p. 33 and refers only to mathematical objects, symbols, or structures without any reference to the real world. However, PISA only uses a limited number of such
contexts in its surveys and places most value on real-world contexts, which are called ‘extra- mathematical contexts’ OECD, 2003b, p. 33. To solve tasks which use extra-mathematical
contexts, students need to translate the contexts into a mathematical form through the process of mathematization OECD, 2003b.
The extra-mathematical contexts defined by PISA are similar to Roth’s 1996 definition of contexts, which also focuses on the modeling perspective. Roth 1996, p. 491 defined context as “a
real-world phenomenon that can be modeled by mathematical form.” In comparison to Van den Heuvel-Panhuizen and the PISA researchers, Roth takes a narrower perspective on contexts, because
he restricts contexts only to real-world phenomena. However, despite this restriction, Roth’s focus on the mathematical modeling of the context is close to the idea of mathematization as used in PISA.
Based on the aforementioned definitions of context, in our study we restricted contexts to situations which provide opportunities for mathematization and are connected to daily life. This
restriction is in line with the aim of PISA to assess students’ abilities to apply mathematics in everyday life. In conclusion, we defined context-based mathematics tasks as tasks situated in real-
world settings which provide elements or information that need to be organized and modeled mathematically.
2.2. Solving context-based mathematics tasks Solving context-based mathematics tasks requires an interplay between the real world and
mathematics Schwarzkopf, 2007, which is often described as a modeling process Maass, 2010 or mathematization OECD, 2003b. The process of modeling begins with a real-world problem,
ends with a real-world solution Maass, 2010 and is considered to be carried out in seven steps Blum Leiss, as cited in Maass, 2010. As the first step, a solver needs to establish a ‘situation
model’ to understand the real-world problem. The situation model is then developed into a ‘real model’ through the process of simplifying and structuring. In the next step, the solver needs to
construct a ‘mathematical model’ by mathematizing the real model. After the mathematical model is established, the solver can carry out mathematical procedures to get a mathematical solution of
the problem. Then, the mathematical solution has to be interpreted and validated in terms of the real-world problem. As the final step, the real-world solution has to be presented in terms of the
real-world situation of the problem.
In PISA, the process required to solve a real-world problem is called ‘mathematization’ OECD, 2003b. This process involves: understanding the problem situated in reality; organizing
the real-world problem according to mathematical concepts and identifying the relevant mathematics; transforming the real-world problem into a mathematical problem which represents
the situation; solving the mathematical problem; and interpreting the mathematical solution in terms of the real situation OECD, 2003b. In general, the stages of PISA’s mathematization are
similar to those of the modeling process. To successfully perform mathematization, a student needs to possess mathematical competences which are related to the cognitive demands of
context-based tasks OECD, 2003b. Concerning the cognitive demands of a context-based task, PISA defines three types of tasks:
a. Reproduction tasks These tasks require recalling mathematical objects and properties, performing routine
procedures, applying standard algorithms, and applying technical skills. b. Connection tasks
These tasks require the integration and connection from different mathematical curriculum strands, or the linking of different representations of a problem. The tasks are non-routine and
ask for transformation between the context and the mathematical world. c. Reflection tasks
These tasks include complex problem situations in which it is not obvious in advance which mathematical procedures have to be carried out.
Regarding students’ performance on context-based tasks, PISA OECD, 2009a found that cognitive demands are crucial aspects of context-based tasks because they are – among other
task characteristics, such as the length of text, the item format, the mathematical content, and the contexts – the most important factors influencing item difficulty.
2.3. Analyzing students’ errors in solving context-based mathematics tasks To analyze students’ difficulties when solving mathematical word problems, Newman 1977,
1983 developed a model which is known as Newman Error Analysis see also Clarkson, 1991; Clements, 1980. Newman proposed five categories of errors based on the process of solving
mathematical word problems, namely errors of reading, comprehension, transformation, process skills, and encoding. To figure out whether Newman’s error categories are also suitable for
analyzing students’ errors in solving context-based tasks which provide information that needs to be organized and modeled mathematically, we compared Newman’s error categories with the
stages of Blum and Leiss’ modeling process as cited in Maass, 2010 and the PISA’s mathematization stages OECD, 2003b.
Table 1. Newman’s error categories and stages in solving context-based mathematics tasks
Newman’s error categories
a
Stages in solving context-based mathematics tasks Stages in Blum and Leiss’ Modeling
b
Stages in PISA’s Mathematization
c
Reading: Error in simple recognition of words and
symbols --
--
Comprehension: Error in understanding the meaning of a
problem Understanding problem by establishing
situational model Understanding
problem situated in reality
-- Establishing real model by simplifying
situational model --
-- --
Organizing real-world problems according to
mathematical concepts and identifying
relevant mathematics
Transformation: Error in transforming a word problem
into an appropriate mathematical problem
Constructing mathematical model by mathematizing real model
Transforming real- world problem into
mathematical problem which represents the
problem situation
Process skills: Error in performing mathematical
procedures Working mathematically to get
mathematical solution Solving mathematical
problems
Encoding: Error in representing the mathematical
solution into acceptable written form
Interpreting mathematical solution in relation to original problem situation
Validating interpreted mathematical solution by checking whether this is
appropriate and reasonable for its purpose Interpreting
mathematical solution in terms of real
situation
-- Communicating the real-world solution
--
a
Newman, 1977, 1983; Clarkson, 1991; Clements, 1980;
b
as cited in Maass, 2010;
c
OECD, 2003b
Table 1 shows that of Newman’s five error categories, only the first category that refers to the technical aspect of reading cannot be matched to a modeling or mathematization stage of the
solution process. The category comprehension errors, which focuses on students’ inability to understand a problem, corresponds to the first stage of the modeling process “understanding
problem by establishing situational model” and to the first phase of the mathematization process “understanding problem situated in reality”. The transformation errors refer to errors in
constructing a mathematical problem or mathematical model of a real-world problem, which is also a stage in the modeling process and in mathematization. Newman’s category of errors in
mathematical procedures relates to the modeling stage of working mathematically and the mathematization stage of solving mathematical problems. Lastly, Newman’s encoding errors
correspond to the final stage of modeling process and mathematization at which the mathematical solution is interpreted in terms of the real-world problem situation. Considering these similarities,
Newman’s error categories can be used to analyze students’ errors in solving context-based mathematics tasks.
2.4. Research questions The CoMTI project aims at improving Indonesian students’ performance in solving context-
based mathematics tasks. To find indications of how to improve this performance, the first CoMTI study looked for explanations for the low scores in the PISA surveys by investigating, on
the basis of Newman’s error categories, the difficulties students have when solving context-based mathematics tasks such as used in the PISA surveys.
Generally expressed our first research question was: 1. What errors do Indonesian students make when solving context-based mathematics tasks?
A further goal of this study was to investigate the students’ errors in connection with the cognitive demands of the tasks and the student performance level. Therefore, our second
research question was: 2. What is the relation between the types of errors, the types of context-based tasks in the
sense of cognitive demands, and the student performance level?
3. Method
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